The deeper question might be, is it necessary ever to put it into the oven at all?

I have some real experience with that question and my conclusion is, unambiguously, NO. The difference in results from stovetop vs oven cooking -- assuming you have a good heavy pot in both cases, and know something of what you're doing, of course -- is nil.

In fact, some checking of old European cooking traditions, where this all came from, revealed that they use both methods interchangeably, and the custom of choosing either oven or stovetop was essentially about what stove facilities were available and convenient. In really ancient times, all this happened in a fireplace anyway. Oven is convenient if you were already running it for some reason, or want to warm the place up in cold weather.

Not that there isn't plenty of cooking dogma out there of course, on this as on everything else, and people ready to rationalize it ad infinitum. But I use stove top unless there's a particular reason to choose the oven.

The concrete advantage is that it takes many times less total energy to maintain a stovetop pot at simmering temperature than to do the same thing in an oven.

The problem with the heat-from-all-sides argument (here I'll put on my applied-scientist hat, which is real) is that meat braising, by definition, cooks water-bearing food by moist heat at atmospheric pressure. Under those conditions, the ambient temperature applied to the roast's surface never exceeds boiling-water temperature, 100 °C (212 °F), even in the oven, unless the pot dries out (bad news anyway). Given a heavy pot with a tight lid, the roast will see more or less the same temperature all around, even on stove-top.

You will notice a bit more browned residue on the pot's inside walls, afterwards, with oven cooking (from splashed liquid drying). But the meat doesn't "know" whether the moist heat surrounding it during cooking was conducted from the sides of the pot or just the bottom, as steam. (Same way vegetables, stuffed dumplings, etc. get cooked by "steaming.")

I would suggest that a true braise could at least slightly exceed atmospheric pressure. I do not believe that a true braise can be achieved stove-top; even a crock pot does better. A true brasiere (old school) has a concave lid, fit tight, and the whole thing went into the fire. Coals were placed on top, so that the entire item was surrounded by heat. The item was elevated to come very close to touching the bottom of the lid and create the smallest space possible for cooking the item.

I'm not saying that the process is adiabatic, but there's a technique for raising the pressure: make a paste of flour and water and completed line the rim. That seal can be maintained without failure for hours in the oven under braise. Heat is applied, so the temp has to rise above the boiling temperature. But that might not even matter.

The rest is about water management, and I think that this is the most important thing. On the stove top, there's an inverse thermocline (this is off the top of my head but it is what I've been thinking for years). The top of the lid is exposed to room temperature, yet heat is applied at the bottom. With this, you get a water cycle, where water condenses on the top and drops down over the item. Since I don't want this to occur, I would want heat completely surrounding the container.

Using the paste, the entire container is under constant temperature and pressure, and that eliminates water cycling. No moisture is lost to the atmosphere, because we're not looking for a reduction here in this braise. So there's no liquid drying effect on the side walls. We want to start with little liquid in the first place, and the liquid volume will increase from the juices released by the item. The item almost looks dried out when you open it. It is kind of fun to do a braise, seal it up, and pray, knowing that you won't see it again til you are about to serve your guests.....